Rank-3 scale theorems: Difference between revisions
Removed redirect to Ternary scale theorems Tag: Removed redirect |
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* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps. | * Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps. | ||
* An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 [[billiard scales]]. | * An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 [[billiard scales]]. | ||
== | == Conjectures == | ||
* Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes. | * Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes. | ||
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[[Category:Rank 3]] | [[Category:Rank 3]] | ||
[[Category:Scale]] | [[Category:Scale]] | ||
[[Category:Articles with open problems]] | |||